# Use dot products to show that the points P(1,7,3), Q(0,7,−1), and R(−1,6,2) are the vertices of a right triangle. At which vertex is the right angle ?

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### 1 Answer

When you have two points of the form (x, y, z) and (x', y', z') the vector connecting them is given by R = (x'-x) i +(y'-y) j + (z'-z) k .

Use this relation to find the vectors for each side, then you can check with the dot product of all possible pairs if you get a zero; that would tell you that those two sides are perpendicular to each other.

From the given points, P(1,7,3), Q(0,7,−1), and R(−1,6,2), two vectors can be drawn as: `stackrel rarr(RP) : (-2, -1, -1)` , and `stackrel rarr(RQ): (-1, -1, 3)`

The dot product of these vectors are :

-2*-1 + -1*-1 + -1*3

=2+1-3=0

Hence point R is a right angle.

Consequently, the points P(1,7,3), Q(0,7,−1), and R(−1,6,2) must be the vertices of a right triangle.